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Chapter 11     Dynamics I:   Mass & Inertia


This chapter introduces dynamics, and describes how mass and inertia connect the worlds of kinematics and dynamics.



Table of Contents

What is dynamics?

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  • Dynamics is the study of forces that tend to cause motion
  • Whereas kinematics considers only motion, irrespective of the forces that cause the motion, dynamics is primarily concerned with force
  • Kinematics can be thought of as one component of dynamics, as depicted in Fig.11.1
  • In studies of human movement, the most important dynamics equation is Newton's second law of motion:


\begin{align} \sum F = ma \end{align}


  • Variables:
    • $F$ :     force
    • $m$ :     mass
    • $a$ :     acceleration


  • The sum symbol $\sum$ is used because there can generally be more than one force acting on a body.




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Figure 11.1.   Depiction of the relation between kinematics and dynamics.




Forward and inverse dynamics

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  • Just as kinematics has forward and inverse problems, so too does dynamics
  • Forward dynamics refers to the calculation of kinematics from known or measured forces
  • Inverse dynamics refers to the calculation of forces from known or measured kinematics
  • In human movement studies, which is more difficult: forward or inverse dynamics?
    • Inverse kinematics is much more difficult than forward kinematics (see previous chapters)
    • Forward dynamics is much more difficult than inverse dynamics.
    • Why?
      • Because once inverse kinematics is solved, it is relatively easy to calculate forces if the body segment parameters are known
      • Human movement often involes contact with external objects like the ground, and these forces can be highly complex
      • One reason is that contact surfaces can be irregular.
        • For example, the ground is not always perfectly flat.
        • Contact surfaces usually do not have precisely known friction characteristics.
        • The angle of the contacting body segment can greatly affect forces; for example, contacting a surface with the bony posterior portion of the heel produces much greater forces than contacting the same surface with the soft inferior portion of the heel
      • Since it is generally very difficult to model and accurately predict contact forces, the forward dynamics problem is usually much more difficult than the inverse problem
  • For these reasons, very few software packages support scientifically accurate forward dynamics calculations




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Figure 11.2.   Depiction of the relation between forward dynamics and inverse dynamics.




Inertia

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  • Inertia represents a body's resistance to changes in velocity
  • Inertia is a physical property of all bodies
  • Many different variables characterize the inertial properties of a body
  • The most common inertial property is mass (discussed below in the mass section)
  • Other inertial properties are discussed below in the body segment parameters section
  • As seen in Newton's second law of motion $\left( \sum F = ma \right)$, inertia connects the worlds of kinematics and dynamics
    • Inertia determines how a given set of forces affects the current velocity (forward dynamics)
    • Equivalently, inertia determines what forces generate an observed change in velocity (inverse dynamics)
    • Therefore, instead of regarding dynamics as "forces cause motions", recognize that forces and accelerations change together, and that inertia represents the physical map between these two worlds

Side-note:

  • The word "inertia" is used more generally to refer to general resistance to change, but in dynamics "inertia" is more specific, meaning resistance to changes in velocity

Mass

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  • Mass is the most common inertial parameter
  • Mass quantifies the amount of matter in a body
  • The unit of mass is kilograms (kg)
  • Mass represents a body's resistance to changes in linear velocity
  • Mass is a scalar quantity
    • This implies that the direction of velocity change is irrelevant
    • In other words, mass resists velocity changes equivalently in all directions
  • Unlike weight, which is a force, mass is independent of gravity
    • A body's mass is the same on Earth as it is in space
    • Weight ($W$) is mass ($m$) multiplied by gravitational acceleration ($g$):     $\left(W = mg\right)$
    • Notice that this definition of weight comes directly from Newton's second law of motion
  • Mass is sufficient to consider in dynamics calculations only when describing point motion
    • Descriping body motion requires other inertial parameters, as discussed below.
    • This means that simple projectile motion can be described using only mass
    • However, describing more complex projectile motion (e.g. skydiving) requires additional inertia parameters

Body segment parameters

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  • Body segement parameters (BSPs) are the inertial properties of individual body segments
  • Key BSPs include:
    • Mass
    • Center of mass (COM)
    • Moments of inertia
    • Principal axes of inertia
  • Center of mass (COM)...
    • ...is the location around which a body segment's mass is evenly distributed
    • ...is the same as the body's centroid (i.e., geometric center) only if mass is evenly distributed throughout the body
    • ...is almost always closer to a body's proximal joint than to its distal joint
  • Moments of inertia...
    • ...represent a body's resistance to changes in angular velocity
    • ...serve the same role as mass in the angular version of Newton's second law of motion: $\sum M = I \alpha$
      • $M$ :     moment of force     (the angular equivalent of force)
      • $I$ :     inertia     (the angular equivalent of mass)
      • $\alpha$ :     angular acceleration
    • ...are not scalar quantities.
    • ...form a tensor quantity called the inertia tensor
      • Unlike mass, which is a scalar quantity, and invariant to the direction of a linear velocity change, the inertia tensor can be highly direction dependent.
      • To see why, consider Fig.11.3 below, which depicts two bodies with equal masses, but different moments of inertia.




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Figure 11.3.   Depiction of the difference between mass and moment of inertia. Body A and Body B have the same mass and same volume (assume that mass density is constant throughout both bodies). Since Body A is horizontally and vertically symmetrical, its moments of inertia about the X and Y axes are equivalent. In contrast, Body B's moment of inertia is much greater about the Y axis than about the X axis; in other words, it is much easier to rotate Body B about the X axis than about the Y axis.




  • Principal axes of inertia:
    • Similar to mass, which represent the point about which mass is evenly distributed, the principal axes of inertia represent the axes about which mass is evenly distributed (see Fig.11.4).
    • These prinipal axes of inertia move with the body (see Fig.11.4)
    • For a perfectly symmetrical object (i.e. a circle in 2D space or a sphere in 3D space), it is impossible to uniquely determine the principal axes of inertia.




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Figure 11.4.   A body's principal axes of inertia move with the body.




Relevance to human movement studies

  • Bones and segments in the human body are geometrically irregular
  • Additionally, body segments tend to have multiple types of tissues (bone, muscle, skin, fat, etc.), so generally have non-constant density
  • Moreover, tissues within a body segment can move. For example, muscle contraction generally changes a muscle's shape, and this can affect the segment's inertial properties.
  • This implies that the inertial properties of real body segments are highly complex
  • Errors in BSP estimates imply that dynamics calculations also have errors (both forward and inverse calculations)
  • Measurement of BSPs is therefore an important area of research




BSP measurement methods

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  • Many different methods have been used to measure BSPs
  • One of the most accurate methods for measureing BSPs is magnetic resonance imaging (MRI)
    • MRI can be used to accurately measure the volumes of different tissues in different parts of the body
    • Since the mass densities of these tissues are known, all BSPs can be calculated
    • However, MRI is expensive, and it can take one hour or more to scan a full body
  • MRI is not practical for most human movement experiments, so other methods are used
  • Four key examples of other methods are:
    • Suspension
    • Water displacement
    • Reaction board
    • Geometric models




Suspension

  • Suspension is a method that can be used to measure the location of an object's center of mass (COM)
  • Since the COM is the point about which mass is evenly distributed, suspending an object from a point (that is not the COM) will cause its COM to rest directly beneath the suspension point
  • Drawing a vertical line across a body from its suspension point will intersect with the COM by definition (Fig.11.5)
    • This identifies only the line along which the COM lies
    • Repeating this process once yields a different line, and the intersection point of the two lines is the COM (Fig.11.5)
  • To improve accuracy, this process can be repeated more than twice
  • Advantages:
    • Simple
    • Accurate
  • Disadvantages:
    • It is not possible to physically separate body segements in living humans
    • Difficult to suspend from specific points without damaging the body
    • Suspension causes within-segment tissues (muscle, skin, fat) to move in the gravity direction, thereby changing the COM



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Figure 11.5.   Depiction of the suspension method.




Water displacement

  • Water displacement is a method that can be used to measure a body's volume
  • It works by placing an isolated body inside a volume of water, and measuring the amount of water that is displaced (Fig.11.6)
  • The volume of the displaced water is equivalent to the body's volume
  • If the average density of the segment is known, the mass can also be calculated.
  • Advantages:
    • Simple
    • Accurate
  • Disadvantages:
    • Not possible to isolate body segments in living humans.
    • Possible to obtain approximate values for distal segments in living humans.
    • Mass estimations for human segments are only approximate (due to multiple tissues) unless tissue density is separately measured (e.g. MRI)



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Figure 11.6.   Depiction of the water displacement method for measuring volume. The same method can be applied to human body segments.




Reaction board

  • The reaction board method is a method that can be used to measure segment mass.
  • A reaction board is firm or rigid bed upon which a human stands, sits, or lies (Fig.11.7).
  • When the reaction board is placed on a force platform, the force platform can be used to measure the horizontal COM position of the entire body
  • If individual segments' COM locations are known, the mass of a displaced segment can be calculated by measuring the change in the whole body's horizontal COM when the segment moves a known distance in the horizontal direction.
  • Advantages:
    • Relatively simple
  • Disadvantages:
    • Requires a force platform
    • Requires known segment COM positions



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Figure 11.7.   Depiction of the reaction board method for measuring segment mass. When the whole body mass is known, and the COM locations of individual segments are known, the mass of a displaced segment can be calculated by measuring the whole body's change in COM.Pataky et al. (2003) Clin. Biomech. 18(4), 364–368. Reproduced with author permission.




Geometric models

  • Geometric models can be used to represent body segments, and to calculate all inertia parameters
  • A given body segment is very complex geometrically, but comparatively simple geometric shapes can be used to approximate the body segment geometry (Fig.11.8)
  • A common geometric model for limb segments is a truncated cone or "canonical frustrum".
  • Advantages:
    • Easy
    • If average mass density is known, all inertia parameters can be calculated precisely for the geometric model
  • Disadvantages:
    • May not be accurate
    • Mass densities are generally different for different people



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Figure 11.8.   Depiction of the geometric model method. Relatively simple geometric shapes can be used to represent body segments. Then the inertial properties of the geometric shapes can be calculated precisely.




Summary

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  • Dynamics is the study of forces that tend to cause motion
  • Dynamics encompasses kinematics
  • Forward dynamics is the process of calculating kinematics from forces, and inverse dynamics is the opposite.
  • For human movement studies, forward dynamics is usually much more difficult than inverse dynamics due to complex force behaviors, especially at contacting surfaces
  • Inertia is the resistance to a change in velocity
  • Inertia links dynamics and kinematics
  • Body segment parameters (BSPs) are the inertial properties of human body segments
  • MRI is one of the most accurate methods for measuring BSPs
  • Many other methods for measuring BSPs also exist